Question

For a steady and incompressible flow, the velocity field \( \vec{V} \) in Cartesian \( (x, y, z) \) coordinate system is given as:
\[ \vec{V} = 5x\hat{i} - Py\hat{j} + 3k\hat{k} \] Here, \( \hat{i}, \hat{j}, \hat{k} \) are unit vectors along \( x, y, z \) directions, respectively and \( P \) is a constant.
Which one of the following options is the correct value of \( P \) that satisfies the conservation of mass for the given velocity field?

• A. 5
• B. -5
• C. 8
• D. 2

Hint:

For incompressible flow, the divergence of the velocity field must always be zero to satisfy the conservation of mass.

Answer 1

The Correct Option is A

For an incompressible flow, the conservation of mass implies that the divergence of the velocity field must be zero. Mathematically, this is represented as: \[ \nabla \cdot \vec{V} = 0 \] The velocity field given is: \[ \vec{V} = 5x\hat{i} - Py\hat{j} + 3k\hat{k} \] Now, calculate the divergence of \( \vec{V} \): \[ \nabla \cdot \vec{V} = \frac{\partial}{\partial x}(5x) + \frac{\partial}{\partial y}(-Py) + \frac{\partial}{\partial z}(3k) \] This simplifies to: \[ \nabla \cdot \vec{V} = 5 + (-P) + 0 = 0 \] Solving for \( P \): \[ 5 - P = 0 \Rightarrow P = 5 \] Thus, the correct value of \( P \) is 5.
Therefore, the correct answer is (A) 5.